Multiple choice

If the area of a circle inscribed in an equilateral triangle is $4\pi cm^{2}$. then what is the area of the triangle?

  1. $\displaystyle 12\sqrt{3}cm^{2}$
  2. $\displaystyle 9\sqrt{3}cm^{2}$
  3. $\displaystyle 8\sqrt{3}cm^{2}$
  4. $\displaystyle 18cm^{2}$
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A Correct answer
Explanation

The radius r of the inscribed circle is found from pi*r^2 = 4*pi, so r = 2. For an equilateral triangle, the inradius r is related to the side length s by r = s / (2*sqrt(3)), so s = 2*sqrt(3)*r = 4*sqrt(3). The area of the triangle is (sqrt(3)/4) * s^2 = (sqrt(3)/4) * (16 * 3) = 12*sqrt(3).

AI explanation

Using the area of a circle formula, the incircle radius is sqrt(4) = 2 cm. The relation between an equilateral triangle's inradius and its side is r = a sqrt(3) / 6, so 2 = a sqrt(3) / 6 yielding a = 12 / sqrt(3) cm. The area of an equilateral triangle is a^2 sqrt(3) / 4, which calculates to (144 / 3) sqrt(3) / 4 = 12 sqrt(3) cm^2.

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